Introduced in 1964 by J.E. Broadwell, this model is the classic example of a discrete velocity gas. A discrete velocity model consists of a collection of gas molecules with velocities 
belonging to some finite set 
of discrete velocity vectors in 
. The molecules collide with each other, obeying specified sets of transformation rules which must satisfy basic conservation laws. The models generate systems of highly coupled semi-linear partial differential equations which approximate the Boltzmann equation, and are particularly useful for studying problems in rarefied gas dynamics, such as Couette flow, Rayleigh flow and shock structure, especially at high Mach number (cf. also Gas dynamics, numerical methods of; Gas flow theory; Gas dynamics, equations of).
In the Broadwell model, each identical molecule of mass 
is allowed to move in space with one of the six unit velocity vectors 
, 
, 
, 
, 
, 
. Gas particles are allowed to collide only in pairs, a realistic assumption for gases which are not too dense, and the collision must obey the usual conservation laws for mass, momentum and kinetic energy. Let 
denote a collision of particles with initial velocities 
and 
and final velocities 
and 
. Conservation of momentum dictates that the only possible collisions are then
In each of the above collisions, both mass and kinetic energy are also preserved.
Letting 
denote the number density of molecules with velocity 
, the Boltzmann equation can be written as
where 
and 
are the rates of gain and loss in 
as a result of collisions. Assuming spherical symmetry and collisional cross section 
, one has, for example,
and
since one-third of the 
and 
collisions yield 
pairs.
References
| [a1] | J.E. Broadwell, "Shock structure in a simple discrete velocity gas" Phys. Fluids , 7 (1964) pp. 1243–1247 |
| [a2] | R. Gatignol, "Théorie cinétique d'un gaz répartition discrète de vitesses" , Springer (1975) |
| [a3] | R. Monaco, L. Preziosi, "Fluid dynamic applications of the discrete Boltzmann equation" , World Sci. (1991) |
| [a4] | T. Platkowski, R. Illner, "Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory" SIAM Review , 30 (2) (1988) pp. 213–255 |
How to Cite This Entry:
Broadwell model. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Broadwell_model&oldid=46164
This article was adapted from an original article by M. Ikle (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
